82 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA (c) Let Jt* be the monoid consisting of the adjoints U* of the operators then Jt* is weakly compact (Problem 9(d)). Show that R(X) and F(^*) are ortho- gonal supplementary vector subspaces of E. (Use (b) to reduce to the case where Jt is commutative.) Deduce that F(^) is stable under Jt. (d) Let (AC()i«*«£n be a finite sequence of vectors in E. For each /, let xt — yi + zt , where yt € R(^) and zt e F(*O. Use (c) to show that there exists U e Jt such that U - zt — 0 (1 <£ i ^ n), and then prove that there exists Ue^f such that VU - yi = yt for 1 ^ / <; TZ. (Consider the space E" and the image ^' of Jf in -$?(E"; E") consisting of the operators (?i, . . . , f«)i-»(!r- fi, . . ., 7*- /„) where Te ^, and use the fact that RMO is a vector subspace.) Deduce that the set P(#i, . . . , xn) of operators WzJt such that W - xt ~ W • yt = yt for 1 ^i^n is not empty, and then that the inter- section of all these sets (where (xi) varies over all finite sequences of vectors in E) is nonempty (Section 12.3, Problem 6(e)). Finally, deduce that the projection P : E ™> R(~O corresponding to the direct sum decomposition E = R(M) © F(^) belongs to ^ and hence to ^, and use Problem 10 to conclude that R(^) and F(*^) are orthogonal supplementary vector subspaces (" Jacobs' theorem "). (e) Show that for each C/e^T there exists Ve^f such that VU - y*=y for all y e R(^) (same method as in (d)). In other words, the image ^ of ^ under the map- ping U\-+ {7|R(~O is a subgroup of the unitary group U(R(~O). Show that the orbits of this subgroup are strongly compact in R(^) (cf. Problem 3(c)) and deduce that ^ is strongly compact in .S?(R(JO; R(^0), (f) Let (en)neZ be an orthonormal basis of E, and let Jt be the weakly compact submonoid of *$ generated by the unitary operator U defined by U ' - en — en+1 for all n e Z. Show that F(^) = E and R(^) « {0}. 12. In the notation of Problem 9, let J( be any submonoid of #, containing IE- For each x e E, let T(J? - x) be the closed convex hull of the orbit of x under ^. (a) Show that T(^f - x) contains a ^-invariant point (cf. Problem 3 (a)). Deduce that, if $(<^) denotes the set of ^-invariant vectors and &(~O the set of x e E such that 0 e T(J( • x\ then every vector in E can be written as the sum of a vector belonging to O(^) and a vector belonging to £l(<Jf). (b) Show that <£(~^) and H(^*) are orthogonal supplementary vector subspaces of E. Use Problem 10 to deduce that <D(~O and Q.(^f) are orthogonal supplementary vector subspaces of E. Deduce that T(J? - x) contains only one -^-invariant point. (c) Deduce from (b) that, for each contraction U e tf and each vector x e E, the limit 1 n-l lim - £ Uk-x n-+oo n fc*= i exists and is invariant under U (" von Neumann's ergodic theorem "). (a) Let E be a vector space over R or C, and let V be a vector space of linear forms on E, endowed with the weak topology. Let A be a weakly closed convex set in E, such that 0 g A. Show that there exists a weakly closed hyperplane H in V passing through 0, such that H is the set of all /e V for which /(JCG) = 0 for some JCG 7* 0 in E, and such that g(xG) |> 0 for all g e A. (Use Problem 1 to reduce to the case where V is finite-dimensional, by observing that there exists a neighborhood of 0 containing a closed subspace W of finite codimension in V, and such that the projection of A on a supplement L of W in V does not meet a neighborhood of 0 in L. Then apply the result of Problem 3(a).)#.